Restriction problem with polynomial partitioning

Restriction problem with polynomial partitioning

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Lingfu Zhang, Princeton University
Fine Hall 110

In harmonic analysis, people are interested in the following problem: up to a constant, for any function, can we control the L_q norm of its Fourier transform restricted to the unit sphere, by the L_p norm of the function itself? The restriction conjecture is about all possible pairs (p, q) such that this statement holds. This type of estimates plays an important role in dispersive evolution equations, and the full conjecture implies the Kakeya set conjecture. Since the 70's, when the problem was explicitly posed by Elias Stein, only the 2 dimensional case has been solved. The best current estimate in higher dimensions was given by Larry Guth, using polynomial partitioning. In this talk we will begin with a review of progresses in this conjecture. Then we will talk about the polynomial methods in incidental geometry, and move on to discuss Guth's approach towards the restriction conjecture, focusing on the 3 dimensional case.